{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "61b25a5b-bb05-4dc1-af35-a31d4587648f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "aaaaa\n"
     ]
    }
   ],
   "source": [
    "x = 5\n",
    "x = \"a\" * x\n",
    "print(x)\n",
    "\n",
    "x = 9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8c320c73-ef6d-4089-9b4a-0e3e3b3de5b8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "aaaaa\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "27"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 5\n",
    "x = \"a\" * x\n",
    "print(x)\n",
    "\n",
    "x = 9\n",
    "x*3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "675a2d5b-45b3-4ffc-8f10-38a6a441a246",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 2, 9]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a=[4,2,9]\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "960bbdfd-18bf-4dc5-a47a-a6032e7f4cfd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.pop():\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "6ac491d0-70eb-4fb2-9cef-5fedfc2c0a82",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 2]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "23b97253-5d44-4699-b646-72ea5645178e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n",
      "hello\n"
     ]
    },
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[31m---------------------------------------------------------------------------\u001b[39m",
      "\u001b[31mKeyboardInterrupt\u001b[39m                         Traceback (most recent call last)",
      "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[10]\u001b[39m\u001b[32m, line 4\u001b[39m\n\u001b[32m      2\u001b[39m \u001b[38;5;28;01mwhile\u001b[39;00m \u001b[38;5;28;01mTrue\u001b[39;00m:\n\u001b[32m      3\u001b[39m     \u001b[38;5;28mprint\u001b[39m(\u001b[33m'\u001b[39m\u001b[33mhello\u001b[39m\u001b[33m'\u001b[39m)\n\u001b[32m----> \u001b[39m\u001b[32m4\u001b[39m     \u001b[43mtime\u001b[49m\u001b[43m.\u001b[49m\u001b[43msleep\u001b[49m\u001b[43m(\u001b[49m\u001b[32;43m3\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "\u001b[31mKeyboardInterrupt\u001b[39m: "
     ]
    }
   ],
   "source": [
    "import time\n",
    "while True:\n",
    "    print('hello')\n",
    "    time.sleep(3)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5694e98b-9c6a-4ac5-bbde-fd64c0105bd4",
   "metadata": {},
   "outputs": [],
   "source": [
    "ii\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8a934aee-481e-4bef-946f-22dbee2fca75",
   "metadata": {},
   "outputs": [],
   "source": [
    "ii\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "cf46eb76-e3b7-4554-b06e-30007a112867",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "done\n"
     ]
    }
   ],
   "source": [
    "x=20\n",
    "print('done')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d25fdfed-5a7e-4c41-a919-2980a97e529c",
   "metadata": {},
   "source": [
    "# 一级标题\n",
    "这是一段普通的段落文本，用于展示 Markdown 的基本文本格式。Markdown 是一种轻量级标记语言，易于学习和使用。\n",
    "\n",
    "## 二级标题\n",
    "### 三级标题\n",
    "#### 四级标题\n",
    "##### 五级标题\n",
    "###### 六级标题\n",
    "\n",
    "## 列表\n",
    "### 无序列表\n",
    "- 列表项 1\n",
    "- 列表项 2\n",
    "  - 子列表项 1\n",
    "  - 子列表项 2\n",
    "\n",
    "### 有序列表\n",
    "1. 第一项\n",
    "2. 第二项\n",
    "3. 第三项\n",
    "\n",
    "## 引用\n",
    "> 这是一个引用示例，通常用于引用他人的话语或段落。\n",
    "> 引用可以有多行。\n",
    "\n",
    "## 代码块\n",
    "### 行内代码\n",
    "使用 `print(\"Hello, World!\")` 可以在 Python 中输出简单的信息。\n",
    "\n",
    "### 代码块def add_numbers(a, b):\n",
    "    return a + b\n",
    "\n",
    "result = add_numbers(3, 5)\n",
    "print(result)\n",
    "## 链接\n",
    "[这是一个指向百度的链接](https://www.baidu.com)\n",
    "\n",
    "## 图片\n",
    "![示例图片](https://picsum.photos/200/300)\n",
    "\n",
    "## 表格\n",
    "| 姓名 | 年龄 | 职业 |\n",
    "| ---- | ---- | ---- |\n",
    "| 张三 | 25   | 工程师 |\n",
    "| 李四 | 30   | 设计师 |\n",
    "\n",
    "## 分隔线\n",
    "---\n",
    "\n",
    "这是分隔线后的内容。\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "1f1f14c0-8690-468a-b6f9-ab85cffa7405",
   "metadata": {},
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid character '，' (U+FF0C) (547727102.py, line 4)",
     "output_type": "error",
     "traceback": [
      "  \u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[13]\u001b[39m\u001b[32m, line 4\u001b[39m\n\u001b[31m    \u001b[39m\u001b[31mMarkdown 支持多种标题层级，方便你组织文档结构。\u001b[39m\n                     ^\n\u001b[31mSyntaxError\u001b[39m\u001b[31m:\u001b[39m invalid character '，' (U+FF0C)\n"
     ]
    }
   ],
   "source": [
    "# Jupyter Notebook 功能展示\n",
    "\n",
    "## 1. 标题层级\n",
    "Markdown 支持多种标题层级，方便你组织文档结构。\n",
    "### 三级标题示例\n",
    "#### 四级标题示例\n",
    "##### 五级标题示例\n",
    "###### 六级标题示例\n",
    "\n",
    "## 2. 文本样式\n",
    "- **粗体文本**：使用两个星号 `**` 包裹文本实现粗体效果。\n",
    "- *斜体文本*：用一个星号 `*` 包裹文本呈现斜体。\n",
    "- ***粗斜体文本***：三个星号 `***` 可以实现粗斜体。\n",
    "- ~~删除线文本~~：两个波浪线 `~~` 可添加删除线。\n",
    "\n",
    "## 3. 列表\n",
    "### 无序列表\n",
    "- 列表项 1\n",
    "- 列表项 2\n",
    "    - 子列表项 1\n",
    "    - 子列表项 2\n",
    "\n",
    "### 有序列表\n",
    "1. 第一项\n",
    "2. 第二项\n",
    "3. 第三项\n",
    "\n",
    "## 4. 引用\n",
    "> 生活不止眼前的苟且，还有诗和远方。\n",
    "> —— 高晓松\n",
    "\n",
    "## 5. 代码展示\n",
    "### 行内代码\n",
    "使用 `print(\"Hello, Jupyter!\")` 可以在 Python 中输出简单的问候语。\n",
    "\n",
    "### 代码块# 定义一个简单的函数\n",
    "def factorial(n):\n",
    "    if n == 0 or n == 1:\n",
    "        return 1\n",
    "    else:\n",
    "        return n * factorial(n - 1)\n",
    "\n",
    "# 计算 5 的阶乘\n",
    "result = factorial(5)\n",
    "print(f\"5 的阶乘是: {result}\")\n",
    "## 6. 链接\n",
    "你可以通过点击 [Jupyter 官方网站](https://jupyter.org/) 了解更多关于 Jupyter 的信息。\n",
    "\n",
    "## 7. 图片\n",
    "![示例图片](https://picsum.photos/800/400)\n",
    "这是一张随机生成的示例图片，用于展示 Markdown 中插入图片的功能。\n",
    "\n",
    "## 8. 表格\n",
    "| 水果名称 | 价格（元/斤） | 产地 |\n",
    "| ---- | ---- | ---- |\n",
    "| 苹果 | 5 | 山东 |\n",
    "| 香蕉 | 3 | 海南 |\n",
    "| 葡萄 | 6 | 新疆 |\n",
    "\n",
    "## 9. 分隔线\n",
    "---\n",
    "分隔线可以将文档内容进行区分，使结构更加清晰。\n",
    "\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a15f543e-1de8-427f-877e-753fefc157f2",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "67ec245f-3ba1-41a5-8941-343d420a991d",
   "metadata": {},
   "source": [
    "<!DOCTYPE html>\n",
    "<html lang=\"en\">\n",
    "\n",
    "<head>\n",
    "    <meta charset=\"UTF-8\">\n",
    "    <meta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\">\n",
    "    <script src=\"https://cdn.tailwindcss.com\"></script>\n",
    "    <link href=\"https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.7.2/css/all.min.css\" rel=\"stylesheet\">\n",
    "    <title>示例网页</title>\n",
    "</head>\n",
    "\n",
    "<body class=\"bg-gray-100 font-sans\">\n",
    "    <!-- 导航栏 -->\n",
    "    <nav class=\"bg-blue-500 p-4\">\n",
    "        <div class=\"container mx-auto flex justify-between items-center\">\n",
    "            <a href=\"#\" class=\"text-white text-xl font-bold\">首页</a>\n",
    "            <ul class=\"flex space-x-4\">\n",
    "                <li><a href=\"#\" class=\"text-white hover:text-gray-200\">关于我们</a></li>\n",
    "                <li><a href=\"#\" class=\"text-white hover:text-gray-200\">服务</a></li>\n",
    "                <li><a href=\"#\" class=\"text-white hover:text-gray-200\">联系我们</a></li>\n",
    "            </ul>\n",
    "        </div>\n",
    "    </nav>\n",
    "    <!-- 主内容 -->\n",
    "    <main class=\"container mx-auto p-8\">\n",
    "        <h1 class=\"text-3xl font-bold text-blue-600 mb-4\">欢迎来到示例网页</h1>\n",
    "        <p class=\"text-gray-700 mb-8\">这是一个使用 HTML 和 Tailwind CSS 构建的示例网页，展示了常见的网页元素。</p>\n",
    "        <!-- 图片 -->\n",
    "        <img src=\"https://picsum.photos/800/400\" alt=\"示例图片\" class=\"rounded-md shadow-md mb-8\">\n",
    "        <!-- 列表 -->\n",
    "        <h2 class=\"text-2xl font-bold text-blue-600 mb-4\">特色列表</h2>\n",
    "        <ul class=\"list-disc list-inside text-gray-700 mb-8\">\n",
    "            <li>简洁易用的设计</li>\n",
    "            <li>响应式布局，适配各种设备</li>\n",
    "            <li>丰富的交互元素</li>\n",
    "        </ul>\n",
    "        <!-- 表格 -->\n",
    "        <h2 class=\"text-2xl font-bold text-blue-600 mb-4\">价格表格</h2>\n",
    "        <table class=\"table-auto w-full border-collapse border border-gray-300 mb-8\">\n",
    "            <thead>\n",
    "                <tr>\n",
    "                    <th class=\"border border-gray-300 p-2 bg-gray-200\">套餐</th>\n",
    "                    <th class=\"border border-gray-300 p-2 bg-gray-200\">价格</th>\n",
    "                    <th class=\"border border-gray-300 p-2 bg-gray-200\">包含内容</th>\n",
    "                </tr>\n",
    "            </thead>\n",
    "            <tbody>\n",
    "                <tr>\n",
    "                    <td class=\"border border-gray-300 p-2\">基础套餐</td>\n",
    "                    <td class=\"border border-gray-300 p-2\">$9.99</td>\n",
    "                    <td class=\"border border-gray-300 p-2\">基本功能使用</td>\n",
    "                </tr>\n",
    "                <tr>\n",
    "                    <td class=\"border border-gray-300 p-2\">高级套餐</td>\n",
    "                    <td class=\"border border-gray-300 p-2\">$19.99</td>\n",
    "                    <td class=\"border border-gray-300 p-2\">所有功能使用及优先支持</td>\n",
    "                </tr>\n",
    "            </tbody>\n",
    "        </table>\n",
    "        <!-- 按钮 -->\n",
    "        <a href=\"#\" class=\"bg-blue-500 text-white py-2 px-4 rounded-md hover:bg-blue-600 transition duration-300\">了解更多</a>\n",
    "    </main>\n",
    "    <!-- 页脚 -->\n",
    "    <footer class=\"bg-gray-800 text-white text-center p-4\">\n",
    "        <p>&copy; 2025 示例网页. 保留所有权利.</p>\n",
    "    </footer>\n",
    "</body>\n",
    "\n",
    "</html>    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6fbd0403-5bbd-4e9d-81a4-332518f731c8",
   "metadata": {},
   "source": [
    "<!DOCTYPE html>\n",
    "<html lang=\"en\">\n",
    "\n",
    "<head>\n",
    "    <meta charset=\"UTF-8\">\n",
    "    <meta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\">\n",
    "    <title>纯 HTML 示例页面</title>\n",
    "</head>\n",
    "\n",
    "<body>\n",
    "    <!-- 网页标题 -->\n",
    "    <h1>欢迎来到纯 HTML 示例页面</h1>\n",
    "\n",
    "    <!-- 段落 -->\n",
    "    <p>这是一个不使用任何 CSS 样式的纯 HTML 示例页面，用于展示 HTML 基础元素的使用。</p>\n",
    "\n",
    "    <!-- 无序列表 -->\n",
    "    <h2>无序列表示例</h2>\n",
    "    <ul>\n",
    "        <li>列表项 1</li>\n",
    "        <li>列表项 2</li>\n",
    "        <li>列表项 3</li>\n",
    "    </ul>\n",
    "\n",
    "    <!-- 有序列表 -->\n",
    "    <h2>有序列表示例</h2>\n",
    "    <ol>\n",
    "        <li>第一项</li>\n",
    "        <li>第二项</li>\n",
    "        <li>第三项</li>\n",
    "    </ol>\n",
    "\n",
    "    <!-- 表格 -->\n",
    "    <h2>表格示例</h2>\n",
    "    <table border=\"1\">\n",
    "        <thead>\n",
    "            <tr>\n",
    "                <th>姓名</th>\n",
    "                <th>年龄</th>\n",
    "                <th>职业</th>\n",
    "            </tr>\n",
    "        </thead>\n",
    "        <tbody>\n",
    "            <tr>\n",
    "                <td>张三</td>\n",
    "                <td>25</td>\n",
    "                <td>工程师</td>\n",
    "            </tr>\n",
    "            <tr>\n",
    "                <td>李四</td>\n",
    "                <td>30</td>\n",
    "                <td>设计师</td>\n",
    "            </tr>\n",
    "        </tbody>\n",
    "    </table>\n",
    "\n",
    "    <!-- 链接 -->\n",
    "    <h2>链接示例</h2>\n",
    "    <p>点击 <a href=\"https://www.example.com\">这里</a> 访问示例网站。</p>\n",
    "\n",
    "    <!-- 图片 -->\n",
    "    <h2>图片示例</h2>\n",
    "    <img src=\"https://picsum.photos/200/300\" alt=\"示例图片\">\n",
    "</body>\n",
    "\n",
    "</html>    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee54cea4-c13c-47e9-b727-de84b8a77857",
   "metadata": {},
   "source": [
    "<!DOCTYPE html>\n",
    "<html lang=\"en\">\n",
    "\n",
    "<head>\n",
    "    <meta charset=\"UTF-8\">\n",
    "    <meta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\">\n",
    "    <title>纯 HTML 示例页面</title>\n",
    "</head>\n",
    "\n",
    "<body>\n",
    "    <!-- 网页标题 -->\n",
    "    <h1>欢迎来到纯 HTML 示例页面</h1>\n",
    "\n",
    "    <!-- 段落 -->\n",
    "    <p>这是一个不使用任何 CSS 样式的纯 HTML 示例页面，用于展示 HTML 基础元素的使用。</p>\n",
    "\n",
    "    <!-- 无序列表 -->\n",
    "    <h2>无序列表示例</h2>\n",
    "    <ul>\n",
    "        <li>列表项 1</li>\n",
    "        <li>列表项 2</li>\n",
    "        <li>列表项 3</li>\n",
    "    </ul>\n",
    "\n",
    "    <!-- 有序列表 -->\n",
    "    <h2>有序列表示例</h2>\n",
    "    <ol>\n",
    "        <li>第一项</li>\n",
    "        <li>第二项</li>\n",
    "        <li>第三项</li>\n",
    "    </ol>\n",
    "\n",
    "    <!-- 表格 -->\n",
    "    <h2>表格示例</h2>\n",
    "    <table border=\"1\">\n",
    "        <thead>\n",
    "            <tr>\n",
    "                <th>姓名</th>\n",
    "                <th>年龄</th>\n",
    "                <th>职业</th>\n",
    "            </tr>\n",
    "        </thead>\n",
    "        <tbody>\n",
    "            <tr>\n",
    "                <td>张三</td>\n",
    "                <td>25</td>\n",
    "                <td>工程师</td>\n",
    "            </tr>\n",
    "            <tr>\n",
    "                <td>李四</td>\n",
    "                <td>30</td>\n",
    "                <td>设计师</td>\n",
    "            </tr>\n",
    "        </tbody>\n",
    "    </table>\n",
    "\n",
    "    <!-- 链接 -->\n",
    "    <h2>链接示例</h2>\n",
    "    <p>点击 <a href=\"https://www.example.com\">这里</a> 访问示例网站。</p>\n",
    "\n",
    "    <!-- 图片 -->\n",
    "    <h2>图片示例</h2>\n",
    "    <img src=\"https://picsum.photos/200/300\" alt=\"示例图片\">\n",
    "</body>\n",
    "\n",
    "</html>    "
   ]
  },
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    "<body>\n",
    "    <table border=\"1\" cellpadding=\"10\" cellspacing=\"0\">\n",
    "        <!-- 表格头部 -->\n",
    "        <thead>\n",
    "            <tr>\n",
    "                <!-- 跨三列的标题 -->\n",
    "                <th colspan=\"3\">产品销售统计</th>\n",
    "            </tr>\n",
    "            <tr>\n",
    "                <!-- 分组表头 -->\n",
    "                <th rowspan=\"2\">产品类别</th>\n",
    "                <th colspan=\"2\">销售情况</th>\n",
    "            </tr>\n",
    "            <tr>\n",
    "                <th>第一季度</th>\n",
    "                <th>第二季度</th>\n",
    "            </tr>\n",
    "        </thead>\n",
    "        <!-- 表格主体 -->\n",
    "        <tbody>\n",
    "            <tr>\n",
    "                <td>电子产品</td>\n",
    "                <td>\n",
    "                    <!-- 嵌套表格 -->\n",
    "                    <table border=\"1\" cellpadding=\"5\" cellspacing=\"0\">\n",
    "                        <tr>\n",
    "                            <th>手机</th>\n",
    "                            <td>500 部</td>\n",
    "                        </tr>\n",
    "                        <tr>\n",
    "                            <th>平板电脑</th>\n",
    "                            <td>300 台</td>\n",
    "                        </tr>\n",
    "                    </table>\n",
    "                </td>\n",
    "                <td>\n",
    "                    <table border=\"1\" cellpadding=\"5\" cellspacing=\"0\">\n",
    "                        <tr>\n",
    "                            <th>手机</th>\n",
    "                            <td>600 部</td>\n",
    "                        </tr>\n",
    "                        <tr>\n",
    "                            <th>平板电脑</th>\n",
    "                            <td>400 台</td>\n",
    "                        </tr>\n",
    "                    </table>\n",
    "                </td>\n",
    "            </tr>\n",
    "            <tr>\n",
    "                <td>家居用品</td>\n",
    "                <td>\n",
    "                    <table border=\"1\" cellpadding=\"5\" cellspacing=\"0\">\n",
    "                        <tr>\n",
    "                            <th>沙发</th>\n",
    "                            <td>100 套</td>\n",
    "                        </tr>\n",
    "                        <tr>\n",
    "                            <th>餐桌椅</th>\n",
    "                            <td>200 套</td>\n",
    "                        </tr>\n",
    "                    </table>\n",
    "                </td>\n",
    "                <td>\n",
    "                    <table border=\"1\" cellpadding=\"5\" cellspacing=\"0\">\n",
    "                        <tr>\n",
    "                            <th>沙发</th>\n",
    "                            <td>120 套</td>\n",
    "                        </tr>\n",
    "                        <tr>\n",
    "                            <th>餐桌椅</th>\n",
    "                            <td>250 套</td>\n",
    "                        </tr>\n",
    "                    </table>\n",
    "                </td>\n",
    "            </tr>\n",
    "        </tbody>\n",
    "        <!-- 表格底部 -->\n",
    "        <tfoot>\n",
    "            <tr>\n",
    "                <td colspan=\"3\">统计时间：2025 年第一季度 - 第二季度</td>\n",
    "            </tr>\n",
    "        </tfoot>\n",
    "    </table>\n",
    "</body>"
   ]
  },
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   "cell_type": "markdown",
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    "\n",
    "\n",
    "牛顿 - 莱布尼茨公式是微积分中的重要定理，它表述为：\n",
    "\\[\n",
    "\\int_{a}^{b} f(x) \\mathrm{d}x = F(b) - F(a)\n",
    "\\]\n",
    "其中 \\(F(x)\\) 是 \\(f(x)\\) 的一个原函数，即 \\(F'(x) = f(x)\\)。\n",
    "\n",
    "对于多元函数 \\(z = f(x,y)\\) 的全微分公式为：\n",
    "\\[\n",
    "\\mathrm{d}z = \\frac{\\partial f}{\\partial x} \\mathrm{d}x + \\frac{\\partial f}{\\partial y} \\mathrm{d}y\n",
    "\\]\n",
    "\n",
    "###{矩阵与向量运算}\n",
    "考虑一个 \\(3\\times3\\) 的矩阵 \\(A\\) 和一个三维向量 \\(\\mathbf{v}\\)：\n",
    "\\[\n",
    "A = \n",
    "\\begin{pmatrix}\n",
    "a_{11} & a_{12} & a_{13} \\\\\n",
    "a_{21} & a_{22} & a_{23} \\\\\n",
    "a_{31} & a_{32} & a_{33}\n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "\\mathbf{v} = \n",
    "\\begin{pmatrix}\n",
    "v_1 \\\\\n",
    "v_2 \\\\\n",
    "v_3\n",
    "\\end{pmatrix}\n",
    "\\]\n",
    "矩阵 \\(A\\) 与向量 \\(\\mathbf{v}\\) 的乘积为：\n",
    "\\[\n",
    "A\\mathbf{v} = \n",
    "\\begin{pmatrix}\n",
    "a_{11}v_1 + a_{12}v_2 + a_{13}v_3 \\\\\n",
    "a_{21}v_1 + a_{22}v_2 + a_{23}v_3 \\\\\n",
    "a_{31}v_1 + a_{32}v_2 + a_{33}v_3\n",
    "\\end{pmatrix}\n",
    "\\]\n",
    "\n",
    "###{级数与求和}\n",
    "幂级数是数学分析中的重要内容，例如函数 \\(e^x\\) 的幂级数展开式为：\n",
    "\\[\n",
    "e^x = \\sum_{n = 0}^{\\infty} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\cdots\n",
    "\\]\n",
    "\n",
    "另外，双重求和在处理二维数据或离散概率分布时很常见，例如：\n",
    "\\[\n",
    "\\sum_{i = 1}^{m} \\sum_{j = 1}^{n} a_{ij}\n",
    "\\]\n",
    "它表示对矩阵 \\((a_{ij})\\) 中所有元素求和。\n",
    "\n",
    "\\subsection{积分变换}\n",
    "傅里叶变换是一种重要的积分变换，对于函数 \\(f(t)\\)，其傅里叶变换定义为：\n",
    "\\[\n",
    "\\mathcal{F}\\{f(t)\\} = F(\\omega) = \\int_{-\\infty}^{\\infty} f(t) e^{-i\\omega t} \\mathrm{d}t\n",
    "\\]\n",
    "而其逆变换为：\n",
    "\\[\n",
    "\\mathcal{F}^{-1}\\{F(\\omega)\\} = f(t) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} F(\\omega) e^{i\\omega t} \\mathrm{d}\\omega\n",
    "\\]\n",
    "\n",
    "\\end{document}    "
   ]
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   "source": [
    "# 复杂数学公式示例\n",
    "\n",
    "## 微积分相关公式\n",
    "牛顿 - 莱布尼茨公式是微积分中的重要定理，它表述为：\n",
    "$$\n",
    "\\int_{a}^{b} f(x) \\mathrm{d}x = F(b) - F(a)\n",
    "$$\n",
    "其中 \\(F(x)\\) 是 \\(f(x)\\) 的一个原函数，即 \\(F'(x) = f(x)\\)。\n",
    "\n",
    "对于多元函数 \\(z = f(x,y)\\) 的全微分公式为：\n",
    "$$\n",
    "\\mathrm{d}z = \\frac{\\partial f}{\\partial x} \\mathrm{d}x + \\frac{\\partial f}{\\partial y} \\mathrm{d}y\n",
    "$$\n",
    "\n",
    "## 矩阵与向量运算\n",
    "考虑一个 \\(3\\times3\\) 的矩阵 \\(A\\) 和一个三维向量 \\(\\mathbf{v}\\)：\n",
    "$$\n",
    "A = \n",
    "\\begin{pmatrix}\n",
    "a_{11} & a_{12} & a_{13} \\\\\n",
    "a_{21} & a_{22} & a_{23} \\\\\n",
    "a_{31} & a_{32} & a_{33}\n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "\\mathbf{v} = \n",
    "\\begin{pmatrix}\n",
    "v_1 \\\\\n",
    "v_2 \\\\\n",
    "v_3\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "矩阵 \\(A\\) 与向量 \\(\\mathbf{v}\\) 的乘积为：\n",
    "$$\n",
    "A\\mathbf{v} = \n",
    "\\begin{pmatrix}\n",
    "a_{11}v_1 + a_{12}v_2 + a_{13}v_3 \\\\\n",
    "a_{21}v_1 + a_{22}v_2 + a_{23}v_3 \\\\\n",
    "a_{31}v_1 + a_{32}v_2 + a_{33}v_3\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "## 级数与求和\n",
    "幂级数是数学分析中的重要内容，例如函数 \\(e^x\\) 的幂级数展开式为：\n",
    "$$\n",
    "e^x = \\sum_{n = 0}^{\\infty} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\cdots\n",
    "$$\n",
    "\n",
    "另外，双重求和在处理二维数据或离散概率分布时很常见，例如：\n",
    "$$\n",
    "\\sum_{i = 1}^{m} \\sum_{j = 1}^{n} a_{ij}\n",
    "$$\n",
    "它表示对矩阵 \\((a_{ij})\\) 中所有元素求和。\n",
    "\n",
    "## 积分变换\n",
    "傅里叶变换是一种重要的积分变换，对于函数 \\(f(t)\\)，其傅里叶变换定义为：\n",
    "$$\n",
    "\\mathcal{F}\\{f(t)\\} = F(\\omega) = \\int_{-\\infty}^{\\infty} f(t) e^{-i\\omega t} \\mathrm{d}t\n",
    "$$\n",
    "而其逆变换为：\n",
    "$$\n",
    "\\mathcal{F}^{-1}\\{F(\\omega)\\} = f(t) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} F(\\omega) e^{i\\omega t} \\mathrm{d}\\omega\n",
    "$$    \n"
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